GIFT  OF 


SUGGESTIONS 

ON  THE 

TEACHING   of  ALGEBRA 

With  Especial  Reference  to  the  Use  of 
DURELL  and  ARNOLD'S  ALGEBRA 


BY 

FLETCHER    DURELL,  Ph.D. 

Head  of  Mathematical  Department,  the  Lawrence- 

viLLE  School;  Author  of  the  Durell  Mathematics 

Series  and  Joint  Author  of  the  Durell- 

Arnold  Mathematics  SERi:Es 


CHARLES  E.  MERRILL  COMPANY 
NEW  YORK  CHICAGO 


^: 


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JltllliliTi 


CONTENTS 


PAGE 

1.  Introductory  Remarks 1 

2.  Typical  Recitation 2 

3.  Second  Part  of  Typical  Recitation 3 

4.  Third  Part  of  Typical  Recitation 4 

5.  Fourth  Part  of  Typical  Recitation 6 

6.  Deficiency  Study 8 

7.  First  Lessons  in  Algebra 8 

8.  Verbal  and  Written  Problems 10 

9.  Graphs 11 

10.  The  Formula 12 

11.  Formation  of  Original  Examples 13 

12.  Extra  Credit  Work 13 

13.  Checking  Results 15 

14.  Self-Reliance   and  Cooperation 17 


468823 


SUGGESTIONS   ON   THE   TEACHING   OF 
ALGEBRA  * 

WITH  ESPECIAL  REFERENCE  TO  THE  USE  OP 

DURELL  AND  ARNOLD'S  ALGEBRAS 

1.  Introductory  Remarks. — The  teaching  of  the  early 
part  of  algebra,  that  is,  the  transition  from  arithmetic  to 
algebra,  has  always  been  the  most  difficult  part  of  elemen- 
tary mathematical  instruction.  This  difficulty  has  been 
increased  in  certain  ways  in  recent  years  by  the  fact  that 
pupils  entering  the  high  school  are  more  and  more  immature, 
and  by  the  many  new  and  attractive  appeals  to  their  atten- 
tion in  other  departments  of  study  as  well  as  in  the  outside 
world.  Also  the  recent  somewhat  radical  changes  made  in 
the  subject  matter  of  first  year  algebra  call  for  some  modifi- 
cations in  methods  of  instruction  in  order  to  meet  the  situa- 
tion in  the  most  effective  way. 

This  pamphlet  is  written  in  order  to  give  suggestions  or  illustrations 
to  such  teachers  as  desire  all  the  help  they  can  get,  from  whatever 
quarter,  in  order  to  get  the  maximum  of  results  when  working  under 
the  new  conditions. 

In  the  present  situation  more  than  ever,  the  important 
thing  to  do  is  to  utilize  to  the  utmost  the  natural  growth 
processes  of  the  ohild^s  mind;  for  a  half  year  at  least  to 
feed  young  pupils  much  extremely  simple  and  easily  appre- 
ciated material  so  that  in  time  and  often  without  serious 
effort  they  unconsciously  grow  into  the  power  of  doing 
much  harder  work,  and  indeed  develop  an  appetite  and 
demand  for  such  work. 

♦Copyright,  1921,  by  Charles  E.  Merrill  Co. 


%  TE>ICHING  OF  ALGEBRA 

Any  suggestions  from  teachers  with  a  view  to  the  betterment  of  the 
methods  here  presented  will  be  welcomed.  ^ 

In  what  follows^  we  discuss  first  the  form  or  organiza- 
tion of  the  recitation  best  suited  to  meet  new  conditions 
and  afterward  the  methods  of  treating  the  different  kind* 
of  subject  matter.  -^  ^^ ,   .^. 

2.  The  Parts  of  a  Tjrpical  Recitation  in  Algebra  are  four: 

I.  Return  and  discussion  of  corrected  written  work 
(if  any)  done  in  the  last  fifteen  minutes  of  the  preceding 
recitation. 

II.  Discussion  of  the  advance  work  for  the  current 
recitation. 

III.  Assignment  and  explanation  of  the  next  lesson  in 
advance. 

IV.  Written  work  during  the  final  ten  or  fifteen  minutes 
of  recitation. 

The  first  five  or  ten  minutes  of  a  recitation  in  algebra 
may  well  be  occupied  by  a  discussion  of  the  written  test 
work  done  by  the  pupils  in  the  last  few  minutes  of  the  pre- 
ceding recitation.  Special  stress  is  laid  on  these  papers, 
because  the  work  has  all  been  done  under  the  teacher^s  eye 
without  aid  from  any  outside  source.  Each  of  these  papers 
has  been  carefully  corrected  in  red  pencil  by  the  teacher^' 
and  at  the  opening  of  the  recitation  is  returned  to  the  pupil. 

Any  one  method  of  calling  attention  to  the  errors,  or 
to  specially  meritorious  points  on  the  papers,  if  used  con- 
tinuously soon  loses  its  force.  Hence  in  discussing  these 
papers,  in  order  to  keep  the  interest  of  pupils  fresh  and 
active,  as  well  as  to  use  methods  which  fit  the  peculiarities 
of  each  set  of  papers,  different  methods  may  be  used  from 
day  to  day.    Among  those  available  are  the  following: 

(1)  Pupils  who  have  correct  solutions,  or  specially  good 
solutions,  may  be  asked  to  copy  these  on  the  blackboard 
from  their  papers,  after  which  the  teacher  may  comment 


TEACHING  OF  ALGEBRA  3 

upon  them,  or  other  pupils  may  ask  questions  concerning 
them; 

(2)  The  teacher  may  solve  the  problems  in  whole  oi 
in  part  on  the  blackboard  (often  time  may  be  saved  bv 
putting  part  of  the  work  on  the  blackboard  before  the 
recitation  begins) ; 

(3)  In  case  an  error,  the  correction  of  which  should 
be  emphasized,  has  been  committed  by  a  number  of  pupils, 
the  teacher  after  correcting  this  error  may  fasten  the  matter 
more  firmly  in  the  minds  of  the  class  by  reading  a  list  of 
those  who  have  made  the  error,  or  by  stating  the  number 
of  those  who  have  done  so,  or  by  asking  all  those  who  have 
solved  the  problem  correctly  to  raise  their  hands.  The 
same  applies  in  presenting  any  particularly  good  method  of 
solution  found  on  the  papers; 

(4)  If  a  given  pupil  has  made  a  mistake  like  cancelling 

the  a's  in  — , —  the  undivided  attention  of  the  class  to  this 

a+x 

error  can  be  obtained  by  writing  the  fraction  on  the  board 
without  mentioning  the  fact  that  an  error  has  been  made, 
and  then  calling  upon  the  pupil  who  has  made  the  mistake 
to  come  forward  and  simplify  the  fraction  as  he  has  done 
on  his  paper; 

(5)  If  the  class  has  shown  anything  like  a  general  weak- 
ness in  working  the  assigned  problems,  it  may  be  well, 
instead  of  reviewing  the  marked  papers  in  any  of  the  above 
ways,  to  send  the  whole  class  to  the  blackboard  for  ten 
minutes'  drill  on  examples  of  the  given  type,  including  per- 
haps one  or  more  of  the  actual  examples  used  in  the  test 
under  discussion. 

3.  In  the  Second  Part  of  the  Tjrpical  Recitation  the 
advance  lesson  which  the  pupils  have  prepared  since  the  last 
recitation  is  next  considered.  Individual  pupils  are  now  called 
upon  to  put  on  the  board  solutions  of  the  problems  in  the 


4  TEACHING  OF  ALGEBRA 

leeson  and  the  proof  of  principles.  While  this  is  being  done 
by  part  of  thu  class,  various  methods  of  instructing  the  rest 
of  the  class  may  be  followed  by  the  teacher.  For  instance, 
if  some  of  the  solutions  being  put  on  the  board  are  difficult 
and  involve  details  which  require  close  attention,  it  is  often 
well  for  the  teacher  to  point  out  these  features  while  the 
solutions  are  being  written,  and  to  encourage  such  pupils 
as  are  at  their  seats  to  ask  questions  concerning  any  diffi- 
culties which  they  have  had  with  these  problems. 

If  the  work  does  not  require  such  close  attention  as 
this,  some  one  pupil  may  be  sent  to  the  board  to  work  some 
newly  assigned  example  involving  typical  principles  of  the 
advance  lesson.  ^^^^ .  ^ 

Or  the  time  may  be  spent  in  oral  6r  sight  drills  or  reviews. 

Oral  drills  may  be  conducted  in  various  ways,  as 

(1)  By  having  the  pupils  who  are  at  their  seats  open  their  books  at 
a  specified  place  and  solve  certain  simple  examples  orally; 

(2)  By  having  the  teacher  write  on  the  blackboard  simple  improvised 
problems,  which  pupils  solve  orally.  ;  One  of  the  advantages  of  this 
method  is  that  the  attention  of  the  class  is  held  more  closely  owing  to 
the  fact  that  pupils  are  curious  to  know  what  kind  of  example  will 
come  next.  Another  advantage  is  that  when  any  weakness  on  any 
point  is  discerned  the  teacher  can  at  once  follow  it  up  by  devising  a 
line  of  examples  adapted  to  remedy  it; 

(3)  The  drill  may  be  a  review  of  definitions,  or  in  having  pupils 
invent  and  put  on  the  board  expressions  illustrating  the  definitions 
(see  §  11),  as 

Ex.  1.  Write  two  simultaneous  equations  where  p  and  q  are  the 
imknowns  and  whose  solution  will  give  the  results  p  =  1  and  q  =  2. 

Ex.  2.  Write  an  algebraic  expression  of  three  termS]  each  of  which 
contains  both  x  and  y,  one  of  the  terms  being  of  the  fourth,  another 
of  the  third,  and  another  of  the  second  degree. 

(4)  Oral  drill  on  verbal  problems.     (See  §  8.) 

4.  The  Third  Period  of  the  Typical  or  Standard  Recita- 
tion consists  of  an  explanation  of  the  principles  or  processes 
involved  in  the  next  lesson  and  in  the  assignment  of  the 


TEACHING  OF  ALGEBRA  5 

lesson.     In  explaining  the  next  lesson  the  exercise  of  great 
discretion  is  necessary  on  the  part  of  the  teacher.    if~the-- 

_^rocessesTnvolved^in  ^le-^vance^  wor4^-axa_such^a&  the  pupil 

should  be  able  to  analyze  and  grasx^y-^without  other  aid  --^ 
than  that  given  in  the  textbook,  no  explanation  at  all  should 
be  given  by  the  teacher.  If  an  explanation  is  deemed 
advisable,  the  development  of  the  new  algebraic  process 
should  be  made  as  far  as  possible  by  the  method  of  question 
and  answer,  and  the  pupils  be  expected  to  supply  all  possible 
details  of  the  work,  with  the  reasons  for  the  same.       cv 

In  all  instruction  in  mathematics,  one  of  the  most  difficult  matters 
to  determine,  if  not  the  most  difficult,  is  how  much  and  what  kind  of 
help  to  give  pupils.  To  supply  too  much  help  pauperizes  them  and 
renders  their  minds  inert;  to  give  too  little  help  often  discourages  them. 
A  good  rule  in  this  matter  is  to  give  much  help,  especially  to  young 
and  immature  pupils,  in  the  early  stages  of  a  subject,  and  then  gradually  *^ 
to  diminish  the  amount  of  aid. 

If  it  is  found  that  pupils  are  forming  the  habit  of  relying  too  much 
on  help  given  by  the  teacher,  and  are  not  reading  or  studying  the 
explanations  jgiven  in  the  textbook,  a  partial  remedy  for  this  is  for  the 
teacher,  instead  of  giving  an  explanation,  to  have  some  pupil  read  aloud 
the  explanation  given  in  the  textbook  and  at  the  same  time  write  the 
steps  of  the  accompanying  process  on  the  board,  and  answer  any  ques- 
tions which  the  teacher  may  ask  concerning  the  same.  Or  all  the  mem- 
bers of  the  class  may  be  required  to  read  silently  the  statements  in  the 
textbook,  after  which  the  teacher  may  ask  them  questions  concerning 
what  they  have  read. 

After  a  method  like  that  of  solving  a  quadratic  equation 
by  completing  the  square  has  been  explained  or  studied, 
new  interest  may  be  aroused  and  the  matter  fixed  in  the 
minds  of  pupils  by  asking  some  one  pupil  to  come  to  the 
blackboard  and  work  an  example  by  the  given  method. 

His  work  is  followed  with  the  closest  attention  by  every  pupil, 
any  mistakes  which  he  may  make  are  quickly  noted,  and  other  pupils 
often  ask  for  the  privilege  of  showing  whether  they  cannot  solve  a  like 
example  without  making  an  error.     Also  the  fact  that  after  an  explana- 


6  TEACHING  OF  ALGEBRA 

tion  of  a  new  process,  pupils  may  be  called  upon  at  once  io  show 
whether  they  understand  it,  naturally  tends  to  keep  the  attention  of 
the  class  more  alert  during  all  explanations. 

5.  The  Fourth  and  Last  Part  of  the  Standard  Recitation, 

as  has  already  been  stated,  usually  consists  of  written  work 
on  paper  by  the  members  of  the  class  at  their  seats.  Some 
of  this  may  be  extra  credit  work  (see  §  12),  and  in  it  all 
every  effort  should  be  made  to  stimulate  pupils  to  form 
the  habit  of  checking  their  work  (see  §  13).  In  tests  of 
this  sort  it  is  also  well  usually  to  make  one  or  two  of  the 
assigned  examples  review  work. 

In  this  connection  again  the  question  naturally  arises  as  to  whether 
the  teacher  should  give  pupils  any  help  in  this  final  period  of  work, 
and  if  so,  how  much.  In  the  early  part  of  the  year's  work,  it  is  the 
writer's  habit  to  allow  pupils,  when  they  are  in  difficulty  in  this  test 
work,  to  raise  the  hand  and  obtain  permission  to  come  to  the  desk. 
If  he  finds  that  their  difficulties  are  such  as  they  cannot  be  expected  to 
overcome,  using  a  red  pencil  he  makes  such  suggestions  on  their  papers 
as  will  make  it  possible  for  pupils  to  continue  their  solutions,  the  red 
pencil  marks  rendering  it  easy  for  him  afterward  to  make  equitable 
deductions  in  grading  their  papers.  Giving  help  in  this  way  also  aids 
the  teacher  in  gaining  knowledge  of  the  mental  peculiarities  and  weak- 
nesses of  individual  pupils.  As  the  class  progresses,  however,  less  and 
less  help  of  this  kind  is  given  and  toward  the  end  of  the  year  and  in  all 
review  tests  none  whatever  is  supplied  till  all  the  papers  have  been 
handed  in,  corrected,  and  returned  to  the  class. 

As  has  already  been  indicated,  special  stress  is  laid  on 
this  written  work  which  has  been  done  in  the  presence  of 
the  teacher,  and  of  which  he  knows  exactly  how  much  is  the 
pupil^s  own.  Hence  especial  care  is  taken  in  correcting 
and  discussing  it,  in  the  manner  already  described  (§2). 

While  the  class  is  doing  this  written  work  at  their  seats, 
an  opportunity  is  afforded  the  teacher  of  grouping  and 
making  a  rapid  appraisement  of  the  papers  handed  in  at 
the  beginning  of  the  recitation  which  contain  the  work  done 


TEACHING  OF  ALGEBRA      _  7 

outside  the  class  in  preparation  for  the  current  recitation. 
If  any  member  of  the  class  has  failed  to  do  this  work  prop- 
erly, he  is  at  once  called  to  the  desk  and  asked  to  state  the 
reason  for  this  failure.  If  his  failure  is  due  either  to  neglect, 
or  to  lack  of  grasp  of  the  subject  matter,  he  is  at  once  assigned 
to  deficiency  study  (§6). 

As  with  other  parts  of  the  recitation,  pains  should  be 
taken  to  introduce  variety  into  this  last  period  and  thus 
prevent  it  from  becoming  monotonous  and  ineffective. 
Thus  on  some  days  insteg^d  of  having  pupils  do  written  work 
at  their  seats,  it  4^  wbU  to  send  the  entire  class  to  the  black- 
board and  drill  them  there  in  some  way.  Thus  a  special--_ 
group  of  examples  may  be  assigned  to  each  pupil.  Or  a 
set  of  examples  may  be  written  or  indicated.on  the  black- 
board  to  be  worked  alike  by  all  the  pupils.  Any  tendency 
to  copy  each  other's  work  will  be  diminished  by  the  fact 
that  the  abler  pupils  will  quickly  distance  the  weaker  ones; 
but  at  times  in  this  work  a  certain  amount  of  co-operation, 
in  which  the  abler  pupils  aid  weaker  ones  in  overcoming 
their  difficulties  is  desirable,  since  a  pupil  often  has  a  clearer 
appreciation  of  the  troubles  of  a  fellow  pupil  than  the 
teacher  has,  and  the  progress  of  the  class  is  much  facilitated 
by  such  co-operation. 

One  way  of  obtaining  tlie  co-operation  spoken  of  is  the  following: 
If  one  or  more  of  the  pupils  finish  all  of  the  assigned  examples  before 
the  other  members  of  the  class,  ask  each  of  these  more  successful  stu- 
dents to  help  someone  who  is  having  unusual  difficulty. 

The  drill  during  the  final  period  of  the  recitation  may  also 
sometimes  be  effectively  varied  by  stopping  the  work  at  the 
blackboard  at  a  certain  point  and  sending  all  the  pupils  to 
their  seats  and  having  each  pupil  work  on  paper,  without 
aid  from  anyone,  the  same  examples  which  have  just  been 
worked  at  the  board. 


8  TEACHING  OF  ALGEBRA 

Variety  and  fresh  stimulus  may  also  be  introduced  into 
the  work  by  using  this  last  or  fourth  period  of  the  recitation 
in  some  other  quite  distinct  way  as  in  a  competitive  game  or 
drill  of  some  sort  between  two  halves  or  different  groups 
into  which  the  class  has  been  divided. 

6.  Deficiency  Study. — In  many  schools  the  final  hour 
or  period  of  the  day \s  work  is  employed  in  giving  extra  instruc- 
tion and  drill  to  those  pupils  who  have  neglected  to  do  the 
assigned  work,  and  also  those,  who,  while  working  well,  are 
naturally  slow  and  have  difficulty  in  mastering  the  subject. 
If  the  spirit  of  the  class  is  good,  pupils  often  voluntarily  go 
to  this  period  of  extra  drill,  called  deficiency  study. 

A  good  form  of  carrying  on  the  drill  in  deficiency  study 
is  to  have  all  of  the  pupils  go  to  the  blackboard  and  work  a 
set  of  examples  which  have  been  written  on  the  board  or 
listed  there  from  the  textbook.  The  teacher  watches  and 
corrects  the  work,  carrying  the  answers  to  the  examples  on 
a  piece  of  paper  in  the  hand.  Solutions  by  the  pupils  are 
erased  as  soon  as  they  have  been  pronounced  correct  by  the 
teacher. 

After  a  pupil  has  completed  the  solutions,  if  he  has  made 
few  errors  he  is  allowed  to  go,  or  is  asked  to  aid  other  pupils. 
If  he  has  shown  weakness  in  his  work,  he  is  required  to  work 
the  same  examples  again,  either  at  the  board,  or  at  his  seat 
on  paper. 

Not  only  current  work,  but  also  back  topics  in  either  algebra  or 
arithmetic,  in  which  members  of  the  class  have  shown  weakness,  may  be 
reviewed  in  deficiency  study.  ^.^ 

7.  The  First  Lessons  in  Algebra. — As  has  been  stated 
in  §  1  of  this  pamphlet,  it  is  increasingly  important  that,  in 
their  first  study  of  algebra,  young  pupils  be  fed  with  much 
easy  work  which  appeals  to  them,  so  that  in  time  they  will 
grow  by  natural  processes  into  the  power  to  do  more  diflS- 


TEACHING   OF  ALGEBRA  9 

cult  work.  The  teacher  who^mshe»^4e  earry  this  plan  into 
practice  will  find  certain  features  in  Durell  and  Arnold^s 
First  Book  in  Algebra  a  distinct  help  in  so  doing.  Some 
of  these  features  may  be  followed  with  little  change;  others 
of  them  need  to  be  modified  under  certain  circumstances 
and  with  some  classes  in  order  to  obtain  maximum  results. 

Each  chapter  in  this  book  is  divided  into  two  parts. 
In  Part  I  of  each  chapter  only  the  simplest  cases  and  appli- 
cations of  a  principle  are  given,  formal  definitions,  abstract 
theory,  and  complicated  applications  being  placed  in  the 
corresponding  Part  II.  None  of  the  Parts  II  are  to  be 
studied  till  the  second  half  year,  after  all  the  Parts  I  have 
been  gone  over. 

Thus  for  example  in  Chapter  I,  the  definition  of  algebra,  of  a 
binomial,  etc.,  are  postponed  to  the  second  part  of  the  chapter.  In 
teaching  this  Part  I,  the  teacher  is  to  be  at  every  pains  to  aid  the  pupil 
in  realizing  that  in  arithmetic  he  has  already  unconsciously  learned  a 
considerable  amount  of  algebra  in  the  form  of  certain  symbols  and 
simple  formulas,  and  in  learning  how  to  extend  this  knowledge. 

In  the  case  of  some  classes  it  will  be  found  advisable 
after  studying  the  Parts  I  up  to  the  subject  of  Simultaneous 
Equations  (that  is,  through  Chapter  X),  to  go  back  to  the 
beginning  of  the  book  and  go  over  the  book  as  a  whole, 
studying  all  of  the  Parts  I  and  II  in  order  as  they  occur. 

The  teacher  may  also  at  times  utilize  the  division  of 
chapters  into  Parts  I  and  II  in  another  way.  Thus,  while 
the  class  as  a  whole  is  going  over  the  Parts  I,  special  prob- 
lems in,  or  sections  of,  the  Parts  II  may  be  assigned  to 
brighter  pupils  as  extra  credit  work.  In  this  way  some 
teachers  have  found  it  possible  to  teach  the  whole  book  to 
the  brighter  part  of  the  class  in  a  half  year,  this  part  of  the 
class  being  able  to  study  some  other  subject  during  the 
remainder  of  the  year  while  the  rest  of  the  class  are  com- 
pleting the  study  of  first  year  algebra. 


10  TEACHING  OF  ALGEBRA 

So  much  arithmetic  is  reviewed  and  covered  in  the  Parts 
I,  that,  if  pupils  are  found  deficient  in  any  special  arith- 
metical process,  it  is  possible  to  stress  this  till  it  is  thoroughly 
understood. 

Still  another  advantage  of  the  method  of  studying  first 
year  algebra  here  presented  is  that  if  any  pupils  leave  high 
school  at  the  end  of  the  first  half  year,  they  will  have  studied 
all  of  the  main  principles  of  a  whole  yearns  work  in  algebra 
before  leaving. 

8.  Verbal  and  Written  Problems. — It  is  more  and  more 
being  recognized  that  the  study  of  the  verbal  problem  is  of  the 
first  importance  as  a  means  of  inculcating  the  spirit  of  algebra 
and  enabling  pupils  to  realize  its  purpose.  Hence,  such 
study  is  far  more  valuable  than  practice  in  intricate  manipu- 
lations of  symbols. 

The  teacher  who  wishes  to  make  the  utmost  use  of  the 
verbal  problem,  will  find  much  material  already  worked  out 
in  Durell  and  Arnold ^s  Algebra,  and  in  such  a  form  that  it 
may  be  readily  modified  or  enlarged  if  this  is  deemed  advis- 
able. In  accordance  with  the  general  method  of  the  book, 
many  simple  verbal  problems  are  given  before  the  more  diffi- 
cult ones  are  introduced.  Though  usually  presented  as  oral 
exercises,  these  examples  may  be  assigned  as  written  work  if 
the  teacher  regards  this  as  preferable  with  any  given  class. 

The  study  of  elementary  verbal  problems  not  only  cul- 
tivates thought  power  and  an  appreciation  of  the  spirit  of 
algebra,  but  also  is  the  best  preparation  for  the  more  diffi- 
cult work  of  solving  written  problems  and  of  devising  and 
applying  formulas.  Hence,  if  pupils  have  trouble  in  solving 
written  or  formula  problems,  it  is  important  that  the  teacher 
devise  and  teach  many  simple  verbal  problems  adapted  to 
prepare  for  the  solution  of  more  difficult  ones,  as,  for  instance, 
Ex.  22,  p.  64  prepares  for  Ex.  16,  p.  66;  or  Ex.  15,  p.  88, 
for  Ex.  12,  p.  89. 


TEACHING  OF  ALGEBRA  11 

In  general,  no  better  sight  drill  (see  §  3)  can  be  given 
in  time  available  for  oral  work,  than  that  occupied  in  answer- 
ing such  questions  as  the  following: 

Ex.  1.  What  is  the  area  in  square  inches  of  a  rectangle  x  ft.  long 
and  y  in.  wide? 

Ex.  2.  What  is  the  interest  on  y  dollars  at  6  per  cent  for  i  years? 

9.  Graphs. — It  is  important  in  like  manner  that  when  the 
subject  of  graphs  is  taken  up,  the  pupil  at  first  be  given 
many  simple  graphs  to  construct,  till  he  grasps  the  essential 
principles  involved  and  forms  a  liking  for  the  topic.  As  his 
chief  difficulty  at  the  outset  is  that  of  fixing  on  a  convenient 
numerical  scale  o\s  each  axis,  a  completed  graph  to  be  used 
as  a  model  may  be  given  at  first.  (See  p.  27.)  In  later 
work,  give  only  a  small  part  of  the  graph  and  ask  the  pupil 
to  complete  it.  (See  Ex.  4,  p.  28.)  Later  still,  give  no  part 
of  the  graph,  but  only  the  axes  marked  with  their  numerical 
scales  and  ask  the  pupil  to  supply  the  entire  graph.  (See 
pp.  29,  68,  92,  128.) 

If  the  pupil  is  taught  graphing  in  this  simple  progressive 
way  during  the  first  half  year,  he  will  acquire  such  con- 
fidence in  his  powers  that  he  will,  with  comparatively  little 
help  and  much  zest,  take  up  the  more  difficult  cases,  where 
he  must  determine  the  numerical  scales,  draw  two  graphs 
on  the  same  diagram  with  two  different  scales  on  the  vertical 
axis,  and  later  determine  whether  the  line,  circle,  or  bar  type 
of  graph  should  be  employed  in  a  given  case. 

Similarly  in  teaching  the  interpretation  of  graphs,  it  is 
advantageous  to  follow  the  same  progressive  plan.  Simple 
and  easily  answered  questions  like  those  on  page  28  should  be 
asked,  till  the  pupils  grow  by  familiarity  and  practice  into 
the  power  of  readily  seeing  a  deeper  and  more  comprehensive 
meaning  in  graphic  forms. 


12  \  TEACHING  OF  ALGEBRA 

10.  The  Formula  owing  to  its  abstract  appearance  is 
less  suggestive  ai^d  attractive  to  the  pupil  than  the  graph. 
Hence,  especial  pains  should  be  vised  in  following  the  rule  to 
make  the  first  lessons  in  its  use  simple  and  clear.  When 
first  asked  to  solve  a  problem  by  the  use  of  the  formula, 
pupils  often  say,  *'  I  can  work  this  example  by  arithmetic  ^' 
and  then  proceed  to  do  so,  neglecting  the  formula.  In  this 
case,  in  order  to  emphasize  the  nature  of  the  formula,  it  is 
useful  (as  when  an  example  like  Ex.  3,  p.  9,  is  to  be  worked) 
to  require  the  pupil  to  tabulate  the  work  in  some  formal  way 
like  the  following: 


Formula     a  =  Iw. 

Given  I  =  32,  and  w  =  15, 

Find  a 

Process       Substituting  for  th^  known  letters  in  the  formula,  we 
obtain  / 

a  =32X15, 

Hence,        a  =480  .*.   480  s^.  in.    A^s. 
I  \ 

So  in  teaching  the  framing  of  formulas,  give  much 
practice  at  first  in  simple  cases,  where  all  of  the  needed 
letters  are  given.  (See  gx.  14,  p.  52;  Ex.  2,  p.  88,  etc.) 
If  abundant  drill  of  this  kind  is  given  in  the  first  half  year, 
the  pupil  will  grow  without  much  effort  into  the  mastery  of 
the  more  advanced  cases,  where  he  must  supply  his  own 
letters  in  formulating  a  given  process,  or  transform  a  formula 
with  reference  to  the  different  letters  in  it,  or  must  convert 
a  rule  into  a  formula,  or  vice  versa,  or  eliminate  a  letter 
between  two  formulas,  or  study  the  relations  "^jDetween  a 
formula  and  its  graph.  \ 

Particular  attention  is  called  to  Ex.  30,  p.  69,  Ex.  19,  p.  215,  and 
similar  examples,  by  which  especially,  when  treated  orally,  the  pupil  is 
given  a  large  amount  of  training  in  the  quick  transformation  of  rules 


TEACHING  OF  ALGEBRA  13 

into  formulas  and  the  reverse  process.  These  examples,  together  with 
much  of  the  drill  in  oral  language  work  (see,  for  instance,  Exercise  31, 
p.  63)  enable  him  to  acquire  the  formula  habit. 

11.  Formation  of  Original  Examples. — It  was  stated  in 
§  8  that  one  of  the  best  methods  of  cultivating  a  pupiVs 
appreciation  of  the  inner  meaning  and  spirit  of  algebra  is 
drill  in  the  solution  of  verbal  problems.  Another  impor- 
tant way  of  developing  this  appreciation  is  that  of  training 
the  pupil  to  devise  an  algebraic  problem  to  meet  a  given 
set  of  conditions.  Examples  of  the  kind  meant  are  given  in 
§  3,  p.  3,  of  this  pamphlet.  Other  similar  examples  sup- 
plied in  the  textbook  are  Ex.  4,  p.  12;  Ex.  2,  p.  18;  Exs. 
5-7,  p.  19;  Exs.  32-35,  p.  55;  Exs.  54-58,  p.  147. 

Work  of  this  kind^whether  given  as  sight  drill  or  written 
work,  has  the  double  advantage  of  being  both  a  review  of 
definitions  and  principles,  and  a  training  in  thought 
power. 

12.  Extra  Credit  Work. — When  most  of  the  members  of 
a  class  are  immature  and  for  a  considerable  time  are  receiv- 
ing such  elementary  instruction  in  the  early  study  of  algebra 
as  has  been  described,  it  is  highly  desirable  to  have  some 
means  of  assigning  at  times  in  addition  to  the  regular  lesson, 
a  certain  amount  of  more  difficult  work  to  be  accomplished 
by  the  abler  members  of  the  class.  This  additional  work 
should  not  interfere  with  the  regular  plan  of  instruction, 
but  rather,  if  possible,  should  improve  it  by  speeding  up 
and  stimulating  all  members  of  the  class  to  do  harder  work 
and  thus  outgrow  the  elementary  stage  as  soon  as  possible. 

One  device  by  which  this  end  may  be  attained  is  the 
utilization  of  the  task  and  bonus  principle  now  common  in 
the  payment  of  workmen.  Thus,  if  a  workman  does  a  normal 
or  average  amount  of  work,  he  is  rated  as  100  per  cent  effi- 
cient. If  he  accomplishes  more  than  this,  he  is  rated  as, 
say,  110  per  cent,  or  150  per  cent  efficient. 


14  TEACHING  OF  ALGEBRA 

For  instance,  if,  in  shovelling  iron  ore  from  cars  the  task  is  40  tons 
per  day,  and  a  workman  should  succeed  in  unloading  48  tons,  he  is 
regarded  as  120  per  cent  efficient  for  the  day  and  receives  increased 
pay  accordingly. 


Similarly,  if  a  pupil  does  what  may  fairly  be  expected  of 
one  in  his  stage  of  development,  we  may  give  him  a  grade 
of  100.  If  he  achieves  more  than  this,  and  does  other  work 
called  extra  credit  work,  we  may  give  him  a  grade  of,  say, 
120,  or  140. 

Thus,  we  may  give  a  written  test  or  examination  in  the  following 
form: 

Ex.  1.  Solvex2-a:(x+5)=12+a;. 

Ex.2.  5x-2(3x+2)=7. 

Ex.  3.  3(a:+l)(x-l)  =3x^t-^. 

Ex.4.  9i/=3+2(l+4y).      \ 

Ex.  5.  (Extra  credit).     24-5(a;2-2)  =  l-(a:-l)(5x~2). 

The  first  four  of  the  above  examples  are  intended  to 
represent  what  the  average  or  normal  pupil  may  be  able  to 
solve  in  the  test  period,  and  the  correct  solution  of  all  of 
these  will  entitle  him  to  a  grade  of  100.  If  he  also  succeeds 
in  solving  Ex.  5,  his  grade  will  be  125. 

The  same  principle  is  applicable  in  assigning  home  work 
to  be  done  in  preparation  of  a  lesson,  or  the  method  may  be 
varied  slightly  by  assigning  say  twelve  problems,  the  solu- 
tion of  any  ten  of  which  entitles  a  pupil  to  a  grade  of  100, 
the  grade  for  the  correct  solution  of  all  being  120. 

The  following  are  among  the  advantages  of  the  above  n^ethod  of 
assigning  and  grading  work.  ^\ 

(1)  It  accelerates  the  mental  growth  of  the  class  as  a  whole  and 
keeps  pupils  from  being  content  with  the  more  simple  work  given  them 
at  the  outset. 

(2)  It  prevents  the  more  gifted  pupils  from  settling  down  to  the 


TEACHING  OF  ALGEBRA  15 

general  level  of  the  class,  and,  in  fact,  tends  to  develop  them  to  the 
utmost. 

(3)  It  prevents  a  poor  pupil  from  being  discouraged  by  having  a 
practically  unattainable  standard  of  perfection  set  before  him.  On 
the  contrary  if  he  has  obtained  a  100  per  cent  mark  for  completing  the 
normal  or  bogey  amount  of  work,  he  cheerfully  attacks  other  problems 
with  the  stimulating  feeling  that  he  can  lose  nothing,  but  may  gain 
much  by  so  doing. 

(4)  The  old  100  or  absolute  system  sometimes  had  the  disadvantage 
that  a  pupil  having  gained  a  mark  of  100  or  something  very  close  to  it, 
came  to  feel  that  he  had  learned  about  all  that  was  to  be  known  about 
a  subject.  Hence,  his  progress  was  checked  and  perhaps  ended.  But 
with  possible  bonus  grades  without  limit  above  100  open  to  him,  end- 
less vistas  of  achievement  are  presented  and  suggested  and  the  pupil 
is  started  out  upon  them. 

(5)  The  method  also  has  certain  important  broad  social  and  eco- 
nomic educational  values?  Thus,  one  obstacle  to  the  satisfactory 
settlement  of  certain  labor  and  other  problems  is  the  narrow  view  of 
efficiency  principles  held  both  by  certain  employers  and  some  workmen, 
and  training  by  the  method  here  suggested  helps  broaden  all  who 
become  familiar  with  it. 

The  teacher  can  carry  further  this  training  of  the  efficiency  intelli- 
gence and  conscience  of  pupils,  by  careful  instruction  in  examples  like 
that  in  §  32,  p.  53;  or  like  Ex.  46,  p.  57;  Ex.  8,  p.  202,  etc. 

In  brief  the  method  may  be  made  a  stimulus  to  both 
weak  and  able  pupils  in  several  ways.  _^ 

13.  Checking  Results. — The  extra  credit  principle  de- 
scribed in  §  12  may  be  made  an  aid  in  overcoming  the  reluc- 
tance which  most  pupils  have  to  check  or  prove  an  answer. 

The  more  complex  and  strenuous  the  modern  business 
world  and  life  becomes,  the  more  important  it  is  that  every 
process  and  detail  of  work  should  be  tested  and  proved  so 
that  it  can  be  absolutely  relied  upon,  no  matter  where  and 
how  it  is  used;  Hence  it  is  increasingly  important  that 
children,  as  a  part  of  their  educational  training,  should  form 
the  habit  of  checking  all  answers.  Yet  children  have  a 
marked  distaste  for  this  process,  and  too  often  merely  regard 


16  TEACHING  OF  ALGEBRA 

it  as  a  whimsical  requirement  on  the  part  of  an  exacting 
teacher,  especially  if  the  answer  is  simple  and  exact  and 
"  looks  right/' 

The  ehmination  bf^about  one-third  of  the  more  technical 
and  abstruse  parts  ofs^st  year  algebra  which  has  been 
made  in  some  recent  syllabi  opens  the  way  for  giving  more 
attention  to  the  important  matter  of  checking  each  process 
and  result.  ^^^^-^^ 

Two  devices  may  be  mentioned  for  overcoming  the  dis- 
taste of  pupils  for  proving  answers,  and  for  carrying  pupils 
along  till  they  fully  realize  the  value  of  doing  so,  and  the 
process  becomes  in  a  measure  easy  and  natural. 

The  first  of  these  devices  is  that  of  making  the  checking 
of  a  problem  a  separate  example  and  giving  the  pupil  the 
same  credit  for  checking  a  process  as  for  the  original  solution. 
When  this  is  done  a  sample  test  paper  would  read  as  follows: 

Ex.  1.  Solve  x-2  =  5(x+l)+7. 

Ex.  2.  Check  the  answer  obtained  in  Ex.  1. 

Ex.  3.  Solve  (x+2)(x-3)=x2-7. 

Ex.  4.  Check  the  answer  to  Ex.  3. 

And  so  on  alternately. 

After  pupils  have  thus  been  made  familiar  with  the 
process  of  proving  their  answers,  they  will  in  time  realize 
its  advantage,  and  they  will  voluntarily  use  the  method 
(as  in  an  examination  where  they  are  especially  anxious  to 
get  correct  results)  after  the  above  artificial  stimulus  has 
been  removed. 

The  second  method  of  aiding  pupils  to  form  this  habit 
is  to  begin  with  it  at  the  outset  and  apply  it  to  very  simple 
examples.  If  an  example  is  complicated  the  proving  process 
usually  is  long  and  complex,  and  hence  mistakes  are  likely 
to  be  made  by  the  pupil  during  its  progress.  If  a  dis- 
crepancy thus  arises,  it  is  a  matter  of  considerable  difficulty 


TEACHING  OF  ALGEBRA  17 

for  the  pupil  to  determine  whether  the  error  has  occurred 
in  the  process  of  solving  the  problem  or  in  that  of  checking 
it,  and  the  pupil  comes  to  regard  the  proving  process  as 
merely  an  added  source  of  perplexity.  Hence,  the  pupil 
should  begin  by  checking  many  simple  problems  till  he 
acquires  skill  and  confidence  in  the  application  of  the  process. 

In  this  connection  it  may  be  well  to  state  that  some 
teachers  make  it  a  rule  never  to  tell  a  pupil  whether  an 
answer  is  right  or  wrong,  but  require  pupils  to  test  their 
answers  so  as  to  make  sure  for  themselves  whether  these 
are  correct. 

14.  Seff  Reliance  and  Co-operation. — The  general  plan 
advocated  in  the  preceding  pages,  of  giving  our  present 
immature  pupils  much  s^nple  work  at  the  start  and  thus 
putting  into  action  and  utilizing  the  natural  growth  proc- 
esses of  the  child^s  mind,  also  has  the  advantage  that  when 
pupils  are  treated  thus  they  come  to  work  from  a  higher 
motive,that  is,  more  to  gratify  their  sense  of  mastery  and  ex- 
pansion and  less  to  obtain  good  marks.  They  are  less  likely  to 
^opy  each  other's  work,  or,  indeed,  to  get  an  undue  amount  of 
help  from  any  outside  source;  for  the  pleasure  which  comes 
from  personal  achievement  is  so  great  that  they  want  noth- 
ing to  interfere  with  it,  and  they  often  desire  ever  harder 
work  in  order  to  add  to  this  pleasurable  sense  of  achievement. 

After  this  spirit  has  become  general  in  the  class,  so  that 
a  pupil  is  not  likely  to  accept  aid  except  when  it  is  really 
needed,  it  is  possible  to  allow  and  even  foster  a  certain 
amount  of  co-operation  (see  §  5)  among  the  members  of  a 
class  and  thus  to  utilize  the  remarkable  power  which  some 
pupils  have  of  realizing  the  exact  nature  of  the  stumbling 
blocks  of  other  pupils  and  of  aiding  their  fellow  pupils  to 
overcome  their  difficulties. 


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